LINEAR ALGEBRA EXAM #2 Q&A
How we determine that a 2 × 2 matrix is singular or non-singular in terms of its
determinant? - Answer-if det(A)=0 it is singular (not invertible). if det(A) not =0 it is non
singular (invertible)
How we determine that a 3 × 3 matrix is singular or non-singular in terms of its
determinant? - Answer-if det(A)=0 it is singular (not invertible). if det(A) not =0 it is non
singular (invertible)
definition of the determinant of n × n matrix - Answer-for n≥2 of an nxn matrix [aij] is the
sum of n terms of the form +/-a1jdetAij, with +/- alternating where a11,a12,..,a1n are the
first row entries of A.
definition of the minor of an element aij - Answer-the matrix obtained by deleting the row
and column containing aij.
definition of the cofactor of an element aij - Answer-Aij=(-1)^i+jdet(Mij) where Mij is the
minor of element aij.
the cofactor expansion of the determinant of a n × n matrix A, with n ≥ 2, with respect to
a row or a column - Answer-across the ith row: det(A)=ai1Ai1+ai2Ai2+...+ainAin
down the jth column: det(A)=a1jA1j+a2jA2j+...+anjAnj
relation between det(A) and det(A transpose) - Answer-for square matrices,
det(A)=det(Atranspose)
expression of the determinant of a n × n triangular matrix and its proof - Answer-
det(A)=the product of the entries along the main diagonal (where a is a triangular
matrix)
Proof: cofactor expansion and show that it would be the product of the main diagonal
entries
If a n × n matrix A has two identical rows, what is det(A) - Answer-det(A)=0
If a n × n matrix A has two identical columns, what is det(A) - Answer-det(A)=0
if we exchange two rows or two columns of a n × n det(A), what happens to det(A) -
Answer-det(A)=-det(A)
given a n × n matrix A, prove that ai1Aj1 + ai2Aj2+ . . .+ ainAjn=0 for i is not equal to j. -
Answer-when doing cofactor expansion for the determinant, you will get minor matrices
with identical columns for all terms of the expansion, so the det(A)=0.
, given a n × n matrix A, what is ai1Aj1 + ai2Aj2 + . . . + ainAjn for i = j? - Answer-when i=j
that expression is simply the det(A).
what happens to det(A) of an nxn matrix A when applying row operation I - Answer-
det(A)=-det(A)
what happens to det(A) of an nxn matrix A when applying row operation II - Answer-
det(A)=scalarxdet(A) ---> if you do 1/4R1=R1 you must multiply det(A) by 4, not by 1/4
(multiply by reciprocal of scalar).
what happens to det(A) of an nxn matrix A when applying row operation III - Answer-
det(A)=det(A)
determine the value of the determinant of an elementary matrix of type I - Answer--1
determine the value of the determinant of an elementary matrix of type II - Answer-the
value of the scalar you multiply the matrix by
determine the value of the determinant of an elementary matrix of type III - Answer-1
definition of the adjoint adj(A) of a n × n matrix A - Answer-the transpose of the cofactor
matrix of A
What can you say about the product A adj(A) - Answer-it is equal to a zero matrix
show that if A is a non-singular n × n matrix, then its inverse A−1 is given by 1/det(A) x
adj(A) - Answer-A(adjA)=det(A)I (I being identity matrix or square matrix with diagonal of
1 and all other values 0).
if A is non-singular it follows that det(A) is a nonzero scalar so we may write:
A(1/det(A) x adjA)=I
therefore: A-1=1/det(A) x adj(A)
for which linear systems you can apply the Cramer's rule? - Answer-where there are the
same number of equations as unknowns
ex: 3 equations with 3 variables
What is the Cramer's rule? - Answer-Calculate determinants of the original coefficient
matrix A and of the n matrices resulting from the systematic replacement of a column in
matrix A by the constant matrix B.
Prove the Cramer's rule - Answer-
Cross product - Answer-
Prove Rn is a vector space - Answer-Demonstrate that Rn satisfies the vector space
axioms using matrices
How we determine that a 2 × 2 matrix is singular or non-singular in terms of its
determinant? - Answer-if det(A)=0 it is singular (not invertible). if det(A) not =0 it is non
singular (invertible)
How we determine that a 3 × 3 matrix is singular or non-singular in terms of its
determinant? - Answer-if det(A)=0 it is singular (not invertible). if det(A) not =0 it is non
singular (invertible)
definition of the determinant of n × n matrix - Answer-for n≥2 of an nxn matrix [aij] is the
sum of n terms of the form +/-a1jdetAij, with +/- alternating where a11,a12,..,a1n are the
first row entries of A.
definition of the minor of an element aij - Answer-the matrix obtained by deleting the row
and column containing aij.
definition of the cofactor of an element aij - Answer-Aij=(-1)^i+jdet(Mij) where Mij is the
minor of element aij.
the cofactor expansion of the determinant of a n × n matrix A, with n ≥ 2, with respect to
a row or a column - Answer-across the ith row: det(A)=ai1Ai1+ai2Ai2+...+ainAin
down the jth column: det(A)=a1jA1j+a2jA2j+...+anjAnj
relation between det(A) and det(A transpose) - Answer-for square matrices,
det(A)=det(Atranspose)
expression of the determinant of a n × n triangular matrix and its proof - Answer-
det(A)=the product of the entries along the main diagonal (where a is a triangular
matrix)
Proof: cofactor expansion and show that it would be the product of the main diagonal
entries
If a n × n matrix A has two identical rows, what is det(A) - Answer-det(A)=0
If a n × n matrix A has two identical columns, what is det(A) - Answer-det(A)=0
if we exchange two rows or two columns of a n × n det(A), what happens to det(A) -
Answer-det(A)=-det(A)
given a n × n matrix A, prove that ai1Aj1 + ai2Aj2+ . . .+ ainAjn=0 for i is not equal to j. -
Answer-when doing cofactor expansion for the determinant, you will get minor matrices
with identical columns for all terms of the expansion, so the det(A)=0.
, given a n × n matrix A, what is ai1Aj1 + ai2Aj2 + . . . + ainAjn for i = j? - Answer-when i=j
that expression is simply the det(A).
what happens to det(A) of an nxn matrix A when applying row operation I - Answer-
det(A)=-det(A)
what happens to det(A) of an nxn matrix A when applying row operation II - Answer-
det(A)=scalarxdet(A) ---> if you do 1/4R1=R1 you must multiply det(A) by 4, not by 1/4
(multiply by reciprocal of scalar).
what happens to det(A) of an nxn matrix A when applying row operation III - Answer-
det(A)=det(A)
determine the value of the determinant of an elementary matrix of type I - Answer--1
determine the value of the determinant of an elementary matrix of type II - Answer-the
value of the scalar you multiply the matrix by
determine the value of the determinant of an elementary matrix of type III - Answer-1
definition of the adjoint adj(A) of a n × n matrix A - Answer-the transpose of the cofactor
matrix of A
What can you say about the product A adj(A) - Answer-it is equal to a zero matrix
show that if A is a non-singular n × n matrix, then its inverse A−1 is given by 1/det(A) x
adj(A) - Answer-A(adjA)=det(A)I (I being identity matrix or square matrix with diagonal of
1 and all other values 0).
if A is non-singular it follows that det(A) is a nonzero scalar so we may write:
A(1/det(A) x adjA)=I
therefore: A-1=1/det(A) x adj(A)
for which linear systems you can apply the Cramer's rule? - Answer-where there are the
same number of equations as unknowns
ex: 3 equations with 3 variables
What is the Cramer's rule? - Answer-Calculate determinants of the original coefficient
matrix A and of the n matrices resulting from the systematic replacement of a column in
matrix A by the constant matrix B.
Prove the Cramer's rule - Answer-
Cross product - Answer-
Prove Rn is a vector space - Answer-Demonstrate that Rn satisfies the vector space
axioms using matrices
